Let us eliminate for a moment the effect that occurs when intervening tree branches occult the street light. I have seen this twinkling effect on a windy day, while looking at distant lights in distant neighborhoods where trees are plentiful.
Let me give you my best guess. We know that twinkling of starlight is caused by refraction of light from turbulent eddies in the atmosphere. Planets don't twinkle because their images are much larger than the sizes of the turbulent cells, so many of them get averaged together to give the illusion of a constant image. If you look straight up, most of the turbulent cells are about 9 kilometers above our heads and are about 1/2 meter across. I suspect that something like this is operating for the street lights. But if we were to look towards the horizon, we would not be encountering exactly the same kind of turbulent layers as we do looking straight up. In fact, in areas such as Los Angeles where air masses often stagnate in inversion layers, the atmosphere near the Earth can be rather stable. This would diminish the inherent twinkling ability of the atmosphere along lines of sight towards the horizon.
As you look across the landscape towards the horizon, and towards any specific light, the light passes through more and more of these cells the more distant the street light. At the farthest distance, the averaging effect cancels out the twinkling effect, while at the nearer distances, only a few of these turbulent cells have been intercepted, and the twinkling effect would be more pronounced. For street lights very close to you, the angular size of the light compared to the average size of the cell is probably so large that many of these cells cover the image of the street light, and again the average effect is that of constant illumination and no twinkling.